Applications of the
Bayesian-Weibull Distribution in Life Data Analysis

[Editor's Note: This article has been updated since its original publication to reflect a more recent version of the software interface.]

One of the new features of
Weibull++ is
support for Bayesian statistics. The premise of Bayesian statistics is
to incorporate prior knowledge along with a given set of current
observations in order to make statistical inferences. The prior information
could come from operational or observational data, from previous comparable
experiments or from engineering knowledge.

This type of analysis is particularly
useful if there is a lack of current test data and when there is a strong
prior understanding about the parameter of the assumed life model and a
distribution can be used to model the parameter. By incorporating prior
information about a parameter, a posterior distribution for a parameter can
be produced and an adequate estimate of reliability can be obtained.
Weibull++ introduces a new type of distribution, Bayesian-Weibull, that combines
the concepts of Bayesian statistics with the
properties of the Weibull distribution. The
Reliability
Basics section of this HotWire issue introduces the theory behind the
new distribution and provides two examples to illustrate its application.

Example 1

A manufacturer is testing prototypes of a
modified product. The test is terminated at 2000 hrs, with only two failures
observed from a sample size of eighteen.

Because of the lack of failure
data in the prototype testing, the manufacturer decided to use information
gathered from prior tests on this product to increase the confidence in the
results of the prototype testing. Failure analysis on the prototypes
indicated that the same failure mode as that experienced by the previous
design is observed again. The manufacturer hopes, however, that longer life
has been achieved in the revised product (i.e. that the overall
behavior of the distribution is the same but shifted to the right). Prior
tests have yielded the following values for β.

First, in order to fit the data to a
Bayesian-Weibull model, a prior distribution for β needs to be
determined. Based on the prior tests' β values, the prior
distribution for β was determined to be a lognormal distribution with
μ = 0.8235, σ = 0.0943 (obtained by entering the β data
into a Weibull++ Standard Folio and analyzing it based on the RRX
analysis method).

To use the Bayesian-Weibull distribution,
the manufacturer enters the test data set into a Standard Folio, and then on the control panel, chooses Bayesian-Weibull > B-W Lognormal Prior.
After clicking Calculate, the parameters of the prior β distribution
are entered into the input window, as shown next.

After performing the
calculations, the Folio looks like the one shown next.

The reliability at t = 1000 hours, along
with the two-sided confidence bounds, is calculated using the QCP as
follows:

The reliability plot of the
model is shown next.

A comparison of the
probability plots and confidence bounds for the model and
the regular two-parameter model is shown next. The plot shows that the
confidence bounds obtained using the model are tighter and
therefore lead to a more precise analysis.

Example 2

This example illustrates how Bayesian
analysis can be performed using a prior β distribution that describes
the uncertainty related to the β estimation based on a single data
set rather than a history of β values from many previous analyses.

A manufacturer wants to estimate the warranty
period to offer its customers for a new product. The manufacturer performed
a single test on a previous model of the product and obtained the following
months-to-failure data.

The above figure also shows the results of the
parameter estimation using the two-parameter Weibull and the RRX analysis
method.

From this data, a list of β values can
be obtained based on different confidence level values. In Weibull++, this
is done using the General Spreadsheet and the Function Wizard.

The β for different CL values is
calculated and the following table is generated.

The distribution parameters of β can be
obtained based on using the β values obtained above. In Weibull++,
insert a Free-Form (Probit) data sheet and enter the data as follows:

The best fit distribution for β is
found to be the lognormal distribution. The parameters of the distribution
are found to be μ = 0.8473 and σ = 0.1854. This now
constitutes the prior distribution of
β, which can be used as the prior knowledge in subsequent analysis.

The manufacturer performed another test on a
new version of the product. Because of cost considerations, the test was
limited to six units. Therefore, the manufacturer wants to use the prior
knowledge gathered about
β in the reliability analysis of the new product version.

Using the Bayesian-Weibull distribution and
the β prior distribution, the new product test data set is
entered into the Folio as follows.

The QCP can be used to estimate a
warranty duration for the new product that would meet an 85% reliability at
a 90% confidence level. The warranty duration is estimated to be 20.59
months.